A NNALI
DELLA
S CUOLA N ORMALE S UPERIORE DI P ISA Classe di Scienze
ROBERTO A LICANDRO M ATTEO F OCARDI M ARIA S TELLA G ELLI Finite-difference approximation of energies in fracture mechanics Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 4e série, tome 29, no 3 (2000), p. 671-709.
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671-
Finite-Difference Approximation of Energies in Fracture Mechanics
ROBERTO ALICANDRO - MATTEO FOCARDI MARIA STELLA GELLI
Abstract. We provide a variational approximation by finite-difference energies of functionals of the type
defined for u E SBD(Q), which are related to variational models in ’ fracture mechanics for linearly-elastic materials. We perform this approximation in dimension 2 via both discrete and continuous functionals. In the discrete scheme we treat also boundary value problems and we give an extension of the approximation result to dimension 3. Mathematics
Subject
Classification (2000): 49J45
(primary), 49M25,
74R10
(secondary).
1. - Introduction In this paper we provide of functionals of the type
a
variational
approximation by
discrete
energies
Q with normal v and u E B Here -emu = 2 bounded domain of + denotes the symmetric part of the gradient of u, [u ] is the jump of u is the (n - I)-dimensional Hausdorff measure. through K along v and These functionals are related to variational models in fracture mechanics for linearly elastic materials in the framework of Griffith’s theory of brittle fracture
defined for every closed
hypersurface K c
K; Ilgn), where Q C
is
-
a
.
Pervenuto alla Redazione il 10 novembre 1999
e
in forma definitiva il 13
aprile 2000.
672
(see [33]). In this context u represents the displacement field of the body, with SZ as a reference configuration. The volume term in (1.1) represents the bulk energy of the body in the "solid region", where linear elasticity is supposed to hold, JL, À being the Lame constants of the material. The surface term is the energy necessary to produce the fracture, proportional to the crack surface K in the isotropic case and, in general, depending on the normal v to K and on the jump [u]. The weak formulation of the problem leads to functionals of the type
defined on the space SBD(Q) of integrable functions u whose symmetrized distributional derivative Eu is a bounded Radon measure with density Fu with respect to the Lebesgue measure and with singular part concentrated on an (n - I)-dimensional set Ju, on which it is possible to define a normal vu in a weak sense and one-sided traces. The description of continuum models in Fracture Mechanics as variational limits of discrete systems has been the object of recent research (see [17], [19], [20], [15] and [36]). In particular, in [19] an asymptotic analysis has been performed for discrete energies of the form
where Rs is the portion of the lattice sZ’ of step size > 0 contained in S2 and u : Rs - R’ may be interpreted as the displacement of a particle parameterized In this model the energy of the system is obtained by superposition by x E of energies which take into account pairwise interactions, according to the classical theory of crystalline structures. Upon identifying u in (1.3) with the function in L1 constant on each cell of the lattice the asymptotic behaviour of functionals can be studied in the framework of r-convergence of energies defined on L1 (see [25], [23]). A complete theory has been developed when u is scalar-valued; in this case the proper space where the limit energies are defined is the space of SB V functions (see for instance [24]). An important model case
is when
R~
In this
w)
case we
may rewrite He
as
portion of R, and D! u (x) denotes the difference quotient of this type have been studied also in [22] in Functionals (u (x + 8~) - M(~r)). the framework of computer vision. In [22] and, in a general framework, in [19] it has been proved that, if f (t) min{t, 1 } and p is a positive function with where
is
a
suitable
~
=
673 suitable summability and symmetry of the type
properties,
then R,
approximates
functionals
defined for u E SB V (S2), which are formally very similar to that in (1.2). A similar result holds by replacing min{t, 1 { by any increasing function f with > 0 and f (oo) = b -E-oo. /(0) = 0, Following this approach, in order to approximate (1.2), one may think to "symmetrize" the effect of the difference quotient by considering the family of functionals ..----.......---...
11
-
.......
tend to 0, we obtain as limit a proper subclass of functionals (1.2). coefficients It and k of the limit functionals are related by a fixed ratio. This limitation corresponds to the well-known fact that pairwise interactions produce only particular choices of the Lame constants. To overcome this difficulty we are forced to take into account in the model non-central interactions. The idea underlying this paper is to introduce a suitable discretization of the divergence, call it that takes into account also interactions in directions orthogonal to ~, and to consider functionals of the form
By letting 8
Indeed, the
two
div~u,
with 8 a strictly positive parameter (for more precise definitions see Sections 3 and 7). In Theorem 3.1 we prove that with suitable choices of f, p and 9 we can approximate functionals of type (1.2) in dimension 2 and 3 with arbitrary and (D satisfying some symmetry properties due to the geometry of the lattice. Actually, the general form of the limit functional is the following
with W explicitly given; in particular we may choose W (~u (x ) ) We underline that the energy density of the limit surface term and c is always anisotropic due to the symmetries of the lattices The dependence on [u], vu arises in a natural way from the discretizations chosen and the vectorial framework of the problem. To drop the anisotropy of the limit surface energy we consider as well a continuous version of the approximating functionals (1.5) given by
= 2.
674 where in this
p is
symmetric convolution kernel which corresponds to a polycrystalline approach. By varying f, p and 8, as stated in Theorem 3.8, we obtain
as
case
a
limit functionals of the form
for any choice of positive constants ~c, À and y. This continuous model generalizes the one proposed by E. De Giorgi and studied by M. Gobbino in [31], to approximate the Mumford-Shah functional
defined for U E SBV(Q). The main technical issue of the paper is that, in the proof of both the discrete and the continuous approximation, we cannot reduce to the 1-dimensional case by an integral-geometric approach as in [ 19], [22], [31], due to the presence of the divergence term. For a deeper insight of the techniques used we refer to Sections 4 and 5; we just underline that the proofs of the two approximations (discrete and continuous) are strictly related. Analogously to [19], in Section 7 we treat boundary value problems in the discrete scheme for the 2-dimensional case and a convergence result for such problems is derived (see Proposition 6.3 and Theorem 6.4).
ACKNOWLEDGMENTS. Our attention on this problem was drawn by Andrea Braides, after some remarks by Lev Truskinovsky. We also thank Luigi Ambrosio and Gianni Dal Maso for some useful remarks. This work is part of CNR Research Project "Equazioni Differenziali e Calcolo delle Variazioni". Roberto Alicandro and Maria Stella Gelli gratefully acknowledge the hospitality of Scuola Normale Superiore, Pisa, and Matteo Focardi that of SISSA, Trieste.
2. - Notation and
We denote by (., .) the scalar product in I.I will be the usual euclidean For x, y E JRn, [x, y] denotes the segment between x and y. If a, b E R we write a A b and a V b for the minimum and maximum between a and b, the vector in R2 orthogonal respectively. If ~ = (~1, ~z) E II~2, we denote to ~ defined (-~z, ~ 1). are the families If Q is a bounded open subset of and Borel measure and B is a of open and Borel subsets of Q, respectively. If it norm.
. ’
preliminaries
675 n B). is a Borel set, then the measure It L B is defined as We denote by .en the Lebesgue measure in R’ and by Hk the k-dimensional Hausdorff measure. If B c R’ is a Borel set, we will also use the notation IBI] for L’(B). The notation a.e. stands for almost everywhere with respect to the Lebesgue measure, unless otherwise specified. We use standard notation for
Lebesgue
spaces. We recall also the notion of convergence in measure on the space L 1 (S2; R"). We say that a sequence un converges to u in measure if for every 17 > 0 we = 0. The space L1(Q; when have limn e ~ :un (x) - u(x)]I > endowed with this convergence, is metrizable, an example of metric being
for 2.1. - BV and BD functions Let Q be the
a
complement
bounded open set of JRn. If u E L I (Q; of the Lebesgue set of u, i.e. x §t Su if and
denote only if we
by Su
some .z E If z exists then it is unique and we denote it by E(x). The Su is Lebesgue-negligible and u is a Borel function equal to u .en a.e. in SZ. Moreover, we say that x E Q is a jump point of u, and we denote by Ju the set of all such points for u, if there exist a, b E R" and v E sn-1I such that
for set
a ~ b
and
where jA’
The triplet (a, b, v), uniquely determined by (2.1 ) up to a permutation of (a, b) and a change of sign of v, will be denoted by ( u + (x ) , u - (x ) , vu (x ) ) . Notice that Ju is a Borel subset of Su. We say that u is approximately differentiable at a Lebesgue point x if there exists L
E
such that
If u is approximately differentiable at a Lebesgue point x, then L, uniquely determined by (2.2), will be denoted by Vu(x) and will be called the approximate gradient of u at x.
676
Eventually, given
a
Borel set J C R’ ,
we
if
say that J is
0 and each Ki is a compact subset of a C1 (n - 1) dimenwhere ?-~n-1 (N) sional manifold. Thus, for a 1tn-1-rectifiable set J it is possible to define 1tn-1 a.e. a unitary normal vector field v. =
2.1.1. - BV functions
’
We recall some definitions and basic results on functions with bounded variation. For a detailed study of the properties of these functions we refer to [9] (see also [26], [30]).
DEFINITION 2. l. Let u E L 1 (S2; II~N); we say that u is a function with bounded variation in Q, and we write u E B V (Q; if the distributional derivative Du of u is a N x n matrix-valued measure on Q with finite total variation. If u E BV(Q; R N), then u is approximately differentiable fn a.e. in Q and Ju turns out to be 1tn-1-rectifiable. Let us consider the Lebesgue decomposition of Du with respect to ,Cn, i.e., Du Da u + DS u, where Da u is the absolutely continuous part and DS u is the singular one. The density of Da u with respect to ,Cn coincides ,Cn a.e. with the approximate gradient Vu of u. Define the jump part of Du, DJ u, to be the restriction of DS u to Ju, and the Cantor part, to be the restriction of DS u to Q B Ju, thus we have =
Moreover, it holds b
E
D~u-~u+-u-~®
denotes the matrix whose entries
DEFINITION 2.2. Let u E with bounded variation in Q, and Functionals involved in
we we
write
u E
are
where if a E aibj with 1 i
say that
is
and N and
special function SBV(Q; R N), if Dcu 0. u
a
=
free-discontinuity problems are often not coercive to consider the following wider class (see [24],
SBV(Q; R N), then it is useful [5]).
in
DEFINITION 2.3. Given u E L 1 (S2; II~N), we say that u is a generalized special function with bounded variation in Q, and we write u E G S B V (Q; R N), if g(u) E for every g E such that Og has compact support. Notice that fl L°° (S2; IEBN) f1 L°° (S2; R N). SB V (S2; in S2, and Ju Functions u E differentiable a.e. are IRN) approximately turns out to be ’Hn-1 -rectifiable (see [5]). In the space R N) the following closure and lower semicontinuity theorem holds (see [4], [6]). =
677 and assume that
THEOREM 2.4. Let
If Uh
- u
in
and ’
in Q, theni Moreover
measure
I.
weakly
in
2.1.2. - BD functions We recall some definitions and basic results deformation. For a general treatment of this subject
on we
functions with bounded refer to [8] (see also [12],
[35]). DEFINITION 2.5. Let u E L 1 (S2; bounded deformation in Q, and we write the distributional derivative of u, Eu measure on Q with finite total variation.
say that u is a function with B D (0), if the symmetric part of is a n x n matrix-valued +
we u E
:= 1 (Du D’u),
If
u E
BD(Q), then
turns out to be
u
is
approximately
differentiable
a.e.
in Q and Ju
’}tn-I-rectifiable.
As in the BV case,
where E’u is the
decompose
we can
absolutely
Eu
as
continuous part of Eu with respect to .en with
density
is the restriction of E u to Ju, and E’u is the restriction of ES u to Moreover, E j u turns out to be equal to ,. _ , . , _ , where -
I
, ’.
DEFINITION 2.6. Let u E BD(Q); we say that u is a special function with bounded deformation in Q, and we write u E SBD(Q), if Ecu = 0. °
Before we
set
the and let
stating
Slicing
fix some notation. Let If Moreover, given u : B - R", we define
Theorem
(see [8])
we
678 THEOREM 2.7. Let u E L1(Q; and let ~n ~ be an orthonormal basis Then the following two conditions are equivalent: ...,
Moreover, if u
E
SBD(Q) and ~
E
JRn
B 101 the following properties hold:
The following compactness result in SBD(Q) is due to Bellettini, Coscia and Dal Maso (see [12]) and its proof is based on slicing techniques and on the characterization of SBD(Q) provided by Theorem 2.7. THEOREM 2.8. Let
Then there exists a u E
We state following the
(uj)
subsequence (not relabelled) converging in weakly converge to Eu in L2(Q;
now a
same
lower semicontinuity result in SB D that can be ideas and strategy of the proof of Theorem 2.8.
THEOREM 2.9. Let uj,
for
such that
C
U E
SBD(Q) be such that uj
proved by
in
and
weakly in L2(0) and
Then
In particular, if (2.3) holds for every ~ -~ div u weakly in L2(Q).
-~ u
and
E
{~1, ..., ~n } orthogonal basis in JRn, then
679
Finally,
2.2. -
we
introduce the
following subspace
of
r -convergence
We recall the notion of r-convergence (see [25]). Let (X, d) be a metric of functionals F, : X ---> [0, +00] is said to rspace. A family at U E X, and we write F(u) = to a functional F : X ---> [0, converge if of for positive numbers decreasing to 0 every sequence (8j) r- lim~~o+ F, (u), the following two conditions hold: (i) (lower semicontinuity inequality) for all sequences converging to u in X we have F(u) liming (uj); (ii) (existence of a recovery sequence) there exists a sequence (uj) converging to u in X such that F (u) > lim supj all points We say that F, r-converges to F if F(u) = If we the lower and X F r-limit define u E and that is the of F,. upper r-limits by
F~~
then the conditions (i) and (ii) are equivalent to F’(u) F"(u) F(u). Note that the functions F’ and F" are lower semicontinuous.
respectively,
=
=
will denote by and 1 and on the the lower r-limits r(L)-liminf, r(L)-limsup, space L1 upper 1 endowed with the metric of the L strong convergence and the convergence in In the
measure,
sequel
we
respectively.
The reason for the introduction of this notion is fundamental theorem.
explained by
the
following
THEOREM 2. lo. Let F Fe, and let a compact set K C X exist all s. Then such that infx F, infK Fefor =
=
Moreover, if (uj) is
a
then its limit is a minimum point We refer to
(see also [18]).
sequence such that for F.
converging
[23] for
an
exposition
of the main
limj FSj (uj) properties
of
=
limj infx
r-convergence
680 2.3. - Preliminary lemmas In this section we state and prove some preliminary results, that will be used in the sequel. Let := {~1, ... , ~~ I an orthogonal basis of jRn. Then for any measurable R’ and y E R’ B {0} define function u :
where Notice that
is constant
where I
on
each cell The
following
result
generalizes
Lemma
LEMMA 2.11. Let
then for any compact set K
it
holds
PROOF. For the sake of simplicity we assume B = set K, call I, the double integral in (2.6). the change of variable Ey + s[£]B ~ y we get a
compact
The further
change
{el, ... , en{. With fixed By Fubini theorem and
of variable y - x + E y and Fubini Theorem
thus the conclusion follows by the uniform for strongly converging families in Lfoc REMARK 2.12. Let C, c
6f3
a
family
continuity JRn).
yield
of the translation operator
of sets such that
r-i
681 Then under the hypothesis of the previous lemma, for any compact set K of R’ we can choose y, E Cs such that u in L 1 ( K ; II~n ) . Indeed, by the Mean Value theorem, there exist y, E Cs such that
easily follows from (2.6) and (2.7). This property will be Propositions 4.4 and 5.1. sequel for n 2, ~ E R 2 B101 and ,l3 f ~, ~ 1 ~, we will denote the and [ ~ ] ,~ by and [.1~, respectively.
Then the conclusion used in the proof of In the
operators
=
=
-rectifiable set and define
LEMMA 2.13. Let,
for §
E
where
R’ and
v (x)
is the
unitary normal vector to
PROOF. First note that mula (see [9])
by
J at x.
Fubini theorem and the Generalized Coarea For-
hence
By sets
the very definition of rectifiability there exist 1 Ki of C graphs such that
countably
many compact sub-
682 and
for
hence, first letting s
--~
i ~ j . Thus, by (2.12)
for any
we
have
0 and then M - -E-oo it follows
J compact subset of a C1 graph. Up to an outer approximation with open sets we may assume J open. Furtherly, splitting J into its connected components, we can reduce ourselves to prove the inequality for J connected. For such a J (2.10) follows by an easy com-
Thus, it suffices
to prove
(2.10) for
putation.
D
LEMMA 2.14. Let h :
,,4 (S2) ~ [0,
be a superadditive
function on disjoint
[0,
be a countable
family of Borel functions such that
for every A The
E
proof
of this lemma
can
be found in
[14].
3. - The main result In this section all the results are set in dimension will be given in Section 6. We introduce first a discretization of the for £ > 0 and for any u : R2 --> R2 define
A
generalization
divergence.
Fix
03BE,
E
to
higher
R2 B {0};
683
Starting from this definition we will provide discrete and continuous approximation results for functionals of type (1.2) and (1.6). We underline that this is only one possible definition of discretized divergence that seems to agree with mechanical models of neighbouring atomic interactions. We can give also the following alternative definition
This second definition can be motivated by the fact that from a numerical point of view it gives a more accurate approximation of the divergence as 8 -~ 0, although the centered differences usually have other drawbacks. For the sake of simplicity, in the proofs of Sections 4 and 5, we will assume that the "finite-difference terms" involved in the approximating functionals are defined by (3.1). The arguments we will use can be easily adapted when considering definition (3.2).
3.1. - Discrete Let Q be
a
approximation result
bounded open set of
R 2 and, for c
Let with
~ be
and
for any t > 0. For
where 9 is
a
an
0, define
increasing function,
strictly positive parameter
Then consider the functional
where
>
is such that
and
such that a, b
set
and
defined
as
>
0 exist
684
on
THEOREM 3.1. Let Q be a convex bounded open set L’(0; JR2) to the functional .
with respect where
to
both the
with the function
Then
Fd r-converges
L1(Q; R2)-convergence and the convergence in measure,
defined by
where for
The of
some
proof of the theorem above will be given in Section 4 as a consequence propositions, which deal with lower and upper r-limits separately.
REMARK 3.2. Notice that the surface term
can
be written
explicitly
as
REMARK 3.3. We point out that the assumption Q convex will be used only in the proof of the r- lim sup inequality. This assumption can be weakened (see Remark 4.5).
REMARK 3.4.
Indeed, taking into that
Notice that the domain of Fd is account the assumption on p, an easy
Jae2)
n
SBD2(S2).
computation
shows
685 REMARK 3.5. Note that, by a suitable choice of the discrete function p, the limit functional is isotropic in the volume term, i.e.,
0 elseand p (E ) Choose, for example, to of it is choices where. Moreover, for suitable f and 0, possible approximate functionals of type (3.8) for any strictly positive >i , hi . =
....
By dropping the functional
the
divergence
Gd : L’(Q; II~2)
-~
term in
(3.4) (i.e. 9
[0, +00] defined
and p is
where also the condition
on
=
one can
consider
above and
satisfies
0),
as
as
Then
THEOREM 3.6. Let 0 to the functional
Gd r-converges
L °° ( S2 ;
with respect where
to
both the L 1
(Q; JRz)-convergence and the convergence in measure, I r
p
proof of this theorem can be recovered from the proof of Theorem 3.1, 0 slight modifications. We only remark that the further hypothesis is needed in order to have good coercivity properties of the family G~. REMARK 3.7. Notice that, although the definition of Gd corresponds in 0 0 in (3.5), its r-limit Gd differs from Fd for 0 some sense to taking 8 The
up to
=
=
in the surface term.
3.2. - Continuous
as
approximation result
Let Q be a bounded open set of in the previous section.
R 2 and let f : [0, +00) ~ [0, +00) be
686 For 8
>
0, define
as
where
with
Fc r-converges on L’(0; R 2) the functional
THEOREM 3.8. ___-___-_._ ____ convergence to
_ E _ ____.
with respect to the.
given by
where
Moreover,
Fc converges to Fc pointwise on L’(0; ~2). proof of the theorem above will be consequence
The Sections 4 and 5.
(yi -
REMARK 3.9. Notice that it = a JIR2 p (y) all positive. Moreover, the summability assumption finiteness of such constants. are
of so
on 1/1
propositions
in
that It, h and y easily yields the
REMARK 3.10. We underline that for any positive coefficients /t, X and y, choose f, p and 9 such that the limit functional has the form
we can
to the discrete case, we may the and consider sequence of functionals
Analogously
drop in (3.11)
the -
divergence term
[0,+oo] defined
by
with same
slicing techniques
of
[31 ] it
can
be
and p as above. By applying the proved the following result.
687 THEOREM 3.11. convergence to
on
thefunctional
with respect to theA
given by
i
where
REMARK 3.12. As in the discrete case, the r-limit G’ does not 0.
to Fl for 8
correspond
=
REMARK 3.13. The restriction to L’(Q; R 2) in Theorems 3.1 and 3.8 is technical in order to characterize the r-limit. For a function u in L 1 (S2; R2) B L°° (SZ; ~2), by following the procedure of the proof of Proposition 4.1 below, one can deduce from the finiteness of the r-limits that the one dimensional sections of u belong to Anyway, since condition (i) of Theorem 2.7 is not in general satisfied, one cannot conclude that u E SBD(Q). On the other hand this condition is satisfied if u E BD(Q), so that Theorems 3.1, 3.6, 3.8, 3.11 still hold if we replace L °° ( S2; ~2) by BD(Q).
SBV(Q~).
4. - The discrete
case
In this section we will prove Theorem 3.1. as "localize" the functionals
for any A
E
and
u E
,,4.£ ( S2 ) , where
I
PROPOSITION 4. l. For any
i
In the
sequel
we
need to
688 PROOF. STEP 1. Let
LetEj - 0, We
can
any E E for e E required
first prove the and u
us
in the case f be such that uj +00. In
inequality
(t) u
=
in
(at) A b. measure.
Uj particular, for suppose that liminfj limj 7L2 such that +00. Using this estimate 0, liming j el + e2l, we will deduce that u E SBD(Q) and we will obtain the inequality by proving that, for any ~ E Z~ such that peg) ~ 0, =
To this aim, as in Theorem 4.1 of [19], we will proceed by splitting the lattice Z~ into similar sub-lattices and reducing ourselves to study the limit of functionals defined on one of these sub-lattices. Indeed, fixed EZ2 such that as p (~ ) ~ 0, we split Z~ into an union of disjoint copies
where
where
We
with 1 into
an
where
split as
well the lattice
j
j
union of
disjoint
sub-lattices
as
We confine
now our
attention to the sequence
689 where
and let
(vj)
be the sequence in SBV(r2;
whose components
are
piecewise
affine, uniquely determined by
where
In order to clarify this construction, we note that, in the case ~ = e1, Vj = is piecewise affine along the is the sequence whose direction ei and piecewise constant along the orthogonal direction, for i = 1, 2. It is easy to check that vj still converges to u in measure. Let us fix 17 > 0 and consider A17:= f x E A : dist B A ) > 171. Note that, by construction, for j large we have
component vij
690 and
Then, for j large and for any fixed
In particular by applying a slicing argument and taking into account the notation used in Theorem 2.7, by Fatou Lemma, we get
Note that, even if p (~ 1 ) = 0, taking into account also the divergence term and the second surface term in (4.2), we can obtain an analogue of the inequality (4.3) for §. By the closure and lower semicontinuity Theorem 2.4 we and since u E deduce that uç,y E ~2) we get
for ~ = ~,~.
by assumption p(ei), p(ei1 + e2) # 0, thus (4.4) particular for ~ = el , e2, el + e2. Then by Theorem 2.7, we get that u E SBD(Aq) for any ’1 > 0. Moreover, since the estimate in (4.4) is uniform with respect we conclude that u E SBD(A). back to (4.2), by applying Theorem 2.9 and then letting t7 -~ 0, we get Going holds in
for any
Recall that
691 Note that, using the inequality above with be easily checked that ~ and Lemma 2.14 applied with
Then, by
can
with
we
get
since the argument above is not affected by the choice of the sublattices in which Z2 has been split with respect to ~, we obtain (4.1 ). The thesis follows by summing over ~ E Z~.
Finally,
STEP 2. If f is any increasing positive function satisfying (3.3), we can find two sequences of positive numbers (ai ) and (bi) such that supi ai a, b and f (t) ~: (ai t) A bi for any t > 0. By Step 1 we have that supi bi is finite only if is finite and r (meas)- lim infs-+o =
=
Fd (u)
Then the thesis follows The
inequality
as
above from Lemma 2.14.
will be crucial for the in both the discrete and the continuous case.
following proposition
D
proof
of the
r-limsup
692 PROPOSITION 4.2. Let
PROOF.
Since
Let
us
f (t)
Using
then
the notation of Lemma 2.13, set
b, by Lemma 2.13 there follows
prove that for
a.e.
and for
Let, for instance, § = ~, then using the notation of Theorem 2.7 if . and x we get y + t~, with y E =
Since
we
have that and
follows. and, since Moreover, Jensen’s inequality, Fubini Theorem and (4.5) yield
for , I
Thus
693 Let
us
also prove that
Setting and
(4.8) follows if
we
prove that
Note that
and that
by (4.5)
on
we
have
Thus, by absolute continuity and Jensen’s inequality
Applying
Fubini theorem and then
extending
we
get
Eu to 0 outside SZ
yield
(4.9) follows by the continuity of the translation operator in L2(Q; ~2x2). Of course, using the same argument, we can claim that the analogous inequalities of (4.8), obtained by replacing (~, ~1-) by one among the pairs hold true. Eventually, since f (t) at, by (4.7) and (4.8) we get and
so
,
and the conclusion follows.
694 REMARK 4.3.
Arguing by
as
in the
proof
of
Proposition
4.2
we
infer that the
functionals defined
where
g(t)
:=
(at)
A
b, satisfy the estimate
for any
u E SBD(O). Moreover, by the subadditivity of g and since g(t) by hypothesis,
there holds
Now we are going to prove the r-limsup inequality that concludes the of Theorem 3.1. We will obtain the recovery sequence for u E suitable interpolations of the function u itself.
PROPOSITION 4.4. For any
proof as
U E
PROOF. It suffices to prove the inequality above for u E Up argument we may assume that 0 E Q. Let h E (o, 1 ) and define := for x E := X-10. Notice that Q cc and It is easy to check that in as À ---+ 1 and to a translation
I
.
Then, by the lower semicontinuity of r-lim
it suffices to prove that
for any ~~(0,1). We now generalize an argument used in [22], [14]. consider UÀ extended to 0 outside Notice that for a E have and thus we get
where
is
given by (2.5)
for
Let Sj
-~
EZ2 and ~
E
0 and
Z~
we
695
Then, using the change of variable sj y + a
In
particular, by Proposition
Fix 3
>
we
obtain
4.2 and Remark 4.3, there holds
we
have
for j large
implies
Then, by Remark 2.12, for any j
E
N
and
in . o
Hence, by (4.13) and (4.14), there holds
from which
and
y,
0 and set
By (4.13),
which
-
letting 8
we
-~
infer
0
we
get the conclusion.
we can
choose
such that
696 REMARK 4.5. In the proof of the previous proposition the convexity asThis sumption on S2 is used only to ensure that for any X > 1 S2 cc condition is nedeed in order to justify the existence of some kind of extension of a SBD function outside of S2 with controlled energy. An alternative approach would involve the use of an extension theorem in SBD analogous to the one holding true in SBV (see Lemma 4.11 of [14]). So far, such a result has not been proved, yet. Thus, Proposition 4.4 and Theorem 3.1 can be stated for open sets sharing one of the previous properties.
5. - The continuous In this section
als
F,, ~
case
we
will prove Theorem 3.8. We "localize" the function-
as
for any
with
PROPOSITION 5.1. For any
PROOF. STEP 1.
Let
us
I
first assume be such that In
Let
particular a ~
Fix such
that
for E
a.e.
:I
and such
I1~2 and
"’
Up
to
passing
to a
subsequence We
we
may
assume
that lim
"discretization" argument used in the proof of Proposition 3.38 of [14]. In what follows, when needed, we will consider U j and u extended to 0 outside S2. If we define
we can
write
now
adapt
to our case a
697 where
Fix 8
>
0, then, arguing
Remark 2.12, for any I and in
j
E
N
as
in the
we can
proof
of
find xj
E
Proposition 4.4, by using QE such that
Now we point out that the functionals on the right hand side is of the same type of those defined in (3.4). Hence, up to slight modifications, we can proceed as in the proof of Proposition 4.1 to obtain that u E SBD(Q) and
Finally, recalling that respect to ~ and by Fatou Lemma,
The
expressions
for we
a.e.
by integrating
get
for JL, À, y follow after
a
simple computation.
with
698
we
STEP 2. If f is any, have that r
arguing
as
in
Step 2 of the proof of Proposition 4.1, is finite only if is finite and
’ .
= with supi tt, supi more Lemma 2.14.
ki
=
h and supi yi
For any u
y. The thesis follows
using
once
D
Let us now prove the r-limsup sition 4.2 and Remark 4.3. PROPOSITION 5.2.
=
E
inequality
which
easily
follows
by Propo-
IL~2 ),
PROOF. As usual we can reduce ourselves to prove the inequality for u E For such a u the recovery sequence is provided by the function itself. Indeed, by Proposition 4.2, estimate (4.12) and Fatou lemma, we get
6. -
Convergence of minimum problems in the discrete case 6.1. - A compactness lemma
The following lemma will be crucial to derive the convergence of the minimum problems treated in the next section.
a
LEMMA 6.1. Let f, p, Fd be as in Theorem 3.1; assume in addition that Q is bounded Lipschitz open set. Let (uj) be a sequence in (Q) such that
Then there exists
SBD2(Q).
a
subsequence
converging
in
R2)
to a function
699 PROOF. Without loss of
then the
monotonicity
and
generality
we
subadditivity
may
of
assume
f (t)
=
(at)
f yield
Let
and
then set
Consider the
(piecewise affine)
functions’
defined
on
A
b. Set
700 and in Q. on each elsewhere Notice that triangle a uj vj interpolation of the values of uj on the vertices of the triangle. By direct computation it is easily seen that for any a E Ij and x there holds where
is
=
an
and affine
1)2
E
-
hence, by taking into
Now,
we
provide
an
account
(6.2) and the subadditivity of f,
estimate for
~L1 (J"~ ).
we
get
Let
and note that
By
the
Lipschitz regularity assumption on 0,
and thus
we
it follows
get
Since (Vj) c SBD(Q) is bounded in ~2) and by taking into account (6.3) and (6.4), Theorem 2.8 yields the existence of a subsequence converging u The thesis follows to a function in E SBD2(Q). R 2) noticing that by 0 u in to A. I also is R 2). converging Proposition
(ujk)
Thanks to Lemma 6.1 and by taking into account Theorems 3.1 and 2.10, we have the following convergence result for obstacle problems with Neumann boundary conditions. THEOREM 6.2. Let K be a compact subset of the minimum values
R 2 and let h
E
~2).
Then
converge to the minimum value
Moreover, for any family of minimizers (u,) for (6.5) and for any sequence (8j) of positive numbers converging to 0, there exists a subsequence (not relabeled) Ue. J converging to a minimizer of (6.6).
701
6.2. - Boundary value In this section
problems
deal with boundary value problems for discrete energies. "interior interactions" from those "crossing the a convex set such that 0 E S2, let 11 > 0 and Let cp : 8Q - R 2 E 8Q such that cp is Lipschitz on each connected component of R2 the function §3 : Then define for 1] d i s t (0, we
Following [19], we separate boundary". Let Q c R 2 be denote by the open and p 1, ... , pN
E
{ p 1, ... , by
n aS2. where r >; 0 is such that {rjc} = f1 We remark that §3 E R 2) and J~ The function 3 is a possible extension of w to Q17. Other extensions are possible which, under regularity assumptions, yield the same result. Here we examine this "radial" extension only, for the sake of simplicity. With given u E SBD(Q), let
W1~°°(52,~ ~
then and where y (u) denotes the inner trace of a suitable discretization by
Let
for I
f, p, ,
Fd
be
as
=
u
with respect to
in Theorem 3.1 and
with
assume
UN
Finally,
we
define
in addition that where
I
with
and
7~ underlying
interactions between
inside and the other outside S2
(interactions through
represents that part of the lattice
pairs of points the boundary).
of
Q17
one
702 PROPOSITION 6. 3.
Be r-converges on L
’ (Q; R2) to thefunctionalI
given by
[0,
with respect to both R2) where vaQ is the inner unit normal
-convergence and the convergence in measure, to
aS2 and
the function I>~ :
is
[0,
defined by with for¡
PROOF. Note that if E is
sufficiently small,
for every v
Moreover, the regularity of w and the assumptions
on
=
Let
u E
have
S2 U
Thus,
-
0. as
and
to prove the
Fix 3 > 0, formi
on
0. Then the r- lim inf inequality follows by letting q fix ~~(0,1) and define uf E
in
then
Hence,
L’(Q;
we
f yield
-~ u ~ in measure Let us ~ u in measure on S2, then by Theorem 3.1 and inequalities (6.7) and (6.8), we get
with
E
arguing
r- lim sup inequality,
as
with
in the
proof and
of
it suffices to show that
Proposition 4.4, we
can
find v, of the such that
703 Note that if
with
Then, by the regularity of §3, it
can
be
for E small
proved
we
have
that
hence, by (6.10),
Then, inequality (6.9) follows letting 8
-
0.
D
As a consequence of Lemma 6.1 and Proposition 6.3, convergence result for boundary value problems. THEOREM 6.4. Let K be a compact set Then the minimum values
and let
we
get the following
be as in Proposition 6.3.
converge to the minimum value
Moreover, for any family of minimizers for (6. I 1 ) and for any sequence (8j) exists a subsequence (not relabeled) numbers to there 0, ofpositive converging
converging to a minimizer of (6.12). PROOF. It
easily follows
from Lemma 6.1,
Proposition 6.3
and Theorem 2.10.0
7. - Generalizations
By following dimension can be discrete model in
approach of Section 3, different generalizations to higher proposed. We present here one possible extension of the which provides as well an approximation of energies of
the
type (1.6). For any
orthogonal pair (~, ~ ) E R~ B {0}
and for any
u :
R3 -+
define
704
where ~ x ~ denotes the external product of ~ and Let Q be a bounded open set of R3 and , J
J
1
Then set
for any
const on
and consider the sequence of functionals fined by
with
andf, o
as
in Section 3.
THEOREM 7.1.
Let S2 be
convex.
Then .
the
given by
functional
with respect to both the where
to
L11 (Q; is
R3) -convergence and the convergence in measure, defined by
with for
PROOF. It suffices to proceed as in the proof of Theorem 3.1, by extending all the arguments to dimension 3 and taking into account Lemmas 2.11 and 2.13, D that are stated in any dimension.
705 A
Appendix
In the previous sections in order to study the r-convergence of discrete energies we have identified a function u defined on a lattice with a suitable "piecewise-constant" interpolation, i.e., a function which takes on each cell
of the lattice the value of u in one node of the cell itself. Then, fixed a discretization step length, we treated the convergence (in measure or L1 strong) of discrete functions through this association. This choice is not arbitrarily done. Indeed, the convergence of piecewiseconstant interpolations ensures the convergence of any other "piecewise-affine" ones, whose values on each cell are obtained as a convex combination of the values of the discrete function in the nodes of the cell itself. Actually, the converse result also holds true, as the following proposition shows. PROPOSITION A. I. Let 8 be a positive parameter tending to 0 and let T, n int such that int = 0 if i =A j, be a family of n-simplices in - 0 as 8 ~ 0. Let u£ be diam Rn that and assume also supi Ui We consider a family of functions which are affine on the interior of each simplex the two piecewise constant functions u - E defined on every simplex
(TE )
TE
T£ -
T,.
Ti
by Then
The
) implies
convergence is replaced by
convergence
PROOF. With fixed Bandi E N let to the closed simplex extension of
of
Tl
or
same
local convergence
holds
if L 1
in measure.
Tl R be the unique continuous Tl and let Y;’i’ yjii be two vertices -
such that
and define suppose in addition that of and with the n-simplex homothetic to to see that It is := and let center easy Y homothetyY B ie C 3 inye,iWe will proceed as follows: first we will construct a function v, on Be := and whose distance from u which is affine on the interior of each set tends to 0. Afterwards we will estimate two particular in the convex combinations of ya and ue that will allow us to estimate the oscillation In order to prove as Let vs be defined in x E that If
is constant
we
on
TE
,
B
A, -E- 1 t i . -
B’ 8
int (Bi)
we
observe that
u £ (x - 3 t£ ) .
ratio 3
706 This would be trivial if u were continuous with compact support, and a standard approximation argument for a general u E we have
proved by
which proves (A.1 ). We note that on B’. This point of
is
a
maximum
point of
on
A’ 8
and
a
be Then
can
minimum
gives
Hence
and
By
a
similar argument,
Since - us and z conclude that
are
we
may prove also that
constant on each
simplex
Ti,
and
we
and, finally, by using the following inequalities,
u in L and analogously for Es. If u, - u in or locally in measure, it suffices to repeat the constructions and reasonings above, localizing each integral. For the local conI vergence in measure, one has also to replace the L distance with a distance 0 inducing the convergence in measure on a bounded set.
we
get that z -
707 REMARK A.2. Note that the functions Es, ya do not coincide with the piecewise-constant ones considered in the previous sections. Nevertheless, from the proposition above, one can easily deduce that the convergence (L’, Ll loc or locally in measure) of piecewise-affine functions considered in Lemma 6.1 implies the convergence of the piecewise-constant ones . Indeed, if we consider the piecewise-constant function w, defined on the cell a+[-s/2, E/2)2, a E 8Z2, as us (a) for a given family of functions which are affine on each triangle of the form then it is easy to see UB. Thus, by the previous or locally in measure) implies the proposition the convergence of u, (L1, same convergence of ws. Finally it suffices to note that the piecewise-constant functions considered in Lemma 6.1 can be written as WB(X+TB) with ( cE .
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[email protected] Scuola Normale Superiore Piazza dei Cavalieri, 7 1-56126 Pisa, Italia focardi @cibs.sns.it Via Beirut, 2-4 1-34013 Trieste, Italia
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